AP Statistics
Chapter 8 Outline
8.1 Confidence Intervals
A Confidence Interval for a population parameter is an interval of plausible values for that parameter. It is
constructed in such a way so that, with a chosen degree of confidence, the value of the parameter will be
captured inside the interval.
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic
from a sample is called a point estimate. Ideally, a point estimate is our “best guess” at the value of an
unknown parameter.
Confidence Interval:
 An interval calculated from the data, which has the form:
estimate ± margin of error
statistic ± (critical value) • (standard deviation of statistic)


The margin of error tells how close the estimate tends to be to the unknown parameter in repeated
random sampling.
A confidence level C, the overall success rate of the method for calculating the confidence interval.
That is, in C% of all possible samples, the method would yield an interval that captures the true
parameter value.
Interpreting Confidence Intervals:
Confidence level:
To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this
population will result in an interval that captures the unknown parameter.”
Confidence interval:
To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval
from _____ to _____ captures the actual value of the [population parameter in context].”
Calculating a Confidence Interval:
statistic ± (critical value) • (standard deviation of statistic)
Check the Conditions for A Confidence Interval:
 Random: The data should come from a well-designed random sample or randomized experiment.
 Normal: The sampling distribution of the statistic is approximately Normal.
o For means: population distribution is Normal or n ≥ 30
o For proportions: np ≥ 10 and n(1 – p) ≥ 10.
 Independent: Individual observations are independent. When sampling without replacement, check
10% condition
8.2 Estimating a Population Proportion
Confidence Intervals Four Step Process: **You must do this!!**
State: What parameter do you want to estimate, and at what confidence level?
Plan: Identify the appropriate inference method. Check conditions.
Do: If the conditions are met, perform calculations.
Conclude: Interpret your interval in the context of the problem.
When the standard deviation of a statistic is estimated from data, the results is called the standard error of the
statistic.
One Sample z Interval for a Population Proportion
statistic ± (critical value) • (standard deviation of statistic)
𝑝̂ (1 − 𝑝̂ )
𝑝̂ ± 𝑧 ∗ √
𝑛
Sample Size for a desired Margin of Error:
To determine the sample size n that will yield a level C confidence interval for a population proportion p with a
maximum margin of error ME, solve the following inequality for n:
𝑝̂ (1 − 𝑝̂ )
𝑧∗√
≤ 𝑀𝐸
𝑛
8.3 Estimating a Population Mean
One Sample z Interval for a Population Mean
statistic ± (critical value) • (standard deviation of statistic)
𝑥̅ ± 𝑧 ∗
𝜎
√𝑛
Choosing Sample Size for a Desired Margin of Error When Estimating µ
 Get a reasonable value for the population standard deviation σ from an earlier or pilot study.
 Find the critical value z* from a standard Normal curve for confidence level C.
 Set the expression for the margin of error to be less than or equal to ME and solve for n:
𝜎
𝑧∗
≤ 𝑀𝐸
√𝑛
When 𝝈 is Unknown: Use a t Distribution:
Draw an SRS of size n from a large population that has a Normal distribution with mean µ and standard
deviation σ. The statistic
t
x 
sx n
has the t distribution with degrees of freedom df = n – 1. The statistic will have approximately a tn – 1
distribution as long as the sampling distribution is close to Normal.
Comparing the density curves of the standard Normal distribution and t distributions:

The density curves of the t distributions are
similar in shape to the standard Normal curve.

The spread of the t distributions is a bit greater
than that of the standard Normal distribution.

The t distributions have more probability in the
tails and less in the center than does the standard
Normal.

As the degrees of freedom increase, the t density
curve approaches the standard Normal curve
ever more closely.
Assignment Calendar
1/6 Monday
8.1 # 5-13 odd, 17
1/8 Wednesday
8.2 #27, 31, 33, 35, 37, 41, 43, 47
1/10 Friday
8.3 #55, 57, 59, 63
1/14 Tuesday
8.3 #65, 67, 71, 73
1/16 Thursday
Review
1/21 Tuesday
Test